Peierls-Nabarro energy surfaces and directional mobility of discrete solitons in two-dimensional saturable nonlinear Schrödinger lattices

Annealing approach to root finding

The Newton-Raphson method is a fundamental root-finding technique with numerous applications in physics. In this study, we propose a parameterized variant of the Newton-Raphson method, inspired by principles from physics. Through analytical and empirical validation, we demonstrate that this approach offers increased robustness and faster convergence during root-finding iterations. Furthermore, we establish connections to the Adomian series method and provide a natural interpretation within a series framework. Remarkably, the introduced parameter, akin to a temperature variable, enables an annealing approach. This advancement sets the stage for a fresh exploration of numerical iterative root-finding methodologies.

Spectroscopy and Directed Transport of Topological Solitons in Crystals of Trapped Ions

We study experimentally and theoretically discrete solitons in crystalline structures consisting of several tens of laser-cooled ions confined in a radio frequency trap. Resonantly exciting localized, spectrally gapped vibrational modes of the soliton, a nonlinear mechanism leads to a nonequilibrium steady state of the continuously cooled crystal. We find that the propagation and the escape of the soliton out of its quasi-one-dimensional channel can be described as a thermal activation mechanism. We control the effective temperature of the soliton's collective coordinate by the amplitude of the external excitation. Furthermore, the global trapping potential permits controlling the soliton dynamics and realizing directed transport depending on its topological charge.

Strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media supported by intrinsic gain

We investigate the mobility of nonlinear localized modes in a generalized discrete Ginzburg-Landau-type model, describing a one-dimensional waveguide array in an active Kerr medium with intrinsic, saturable gain and damping. It is shown that exponentially localized, traveling discrete dissipative breather-solitons may exist as stable attractors supported only by intrinsic properties of the medium, i.e., in the absence of any external field or symmetry-breaking perturbations. Through an interplay by the gain and damping effects, the moving soliton may overcome the Peierls-Nabarro barrier, present in the corresponding conservative system, by self-induced time-periodic oscillations of its power (norm) and energy (Hamiltonian), yielding exponential decays to zero with different rates in the forward and backward directions. In certain parameter windows, bistability appears between fast modes with small oscillations and slower, large-oscillation modes. The velocities and the oscillation periods are typically related by lattice commensurability and exhibit period-doubling bifurcations to chaotically "walking" modes under parameter variations. If the model is augmented by intersite Kerr nonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservative system, the existence regime for moving solitons increases considerably, and a richer scenario appears including Hopf bifurcations to incommensurately moving solutions and phase-locking intervals. Stable moving breathers also survive in the presence of weak disorder.

Mobility of solitons in one-dimensional lattices with the cubic-quintic nonlinearity

We investigate mobility regimes for localized modes in the discrete nonlinear Schrödinger (DNLS) equation with the cubic-quintic on-site terms. Using the variational approximation, the largest soliton's total power admitting progressive motion of kicked discrete solitons is predicted by comparing the effective kinetic energy with the respective Peierls-Nabarro (PN) potential barrier. The prediction, for the DNLS model with the cubic-only nonlinearity too, demonstrates a reasonable agreement with numerical findings. A small self-focusing quintic term quickly suppresses the mobility. In the case of the competition between the cubic self-focusing and quintic self-defocusing terms, we identify parameter regions where odd and even fundamental modes exchange their stability, involving intermediate asymmetric modes. In this case, stable solitons can be set in motion by kicking, so as to let them pass the PN barrier. Unstable solitons spontaneously start oscillatory or progressive motion, if they are located, respectively, below or above a mobility threshold. Collisions between moving discrete solitons, at the competing nonlinearities frame, are studied too.

Dragging two-dimensional discrete solitons by moving linear defects

We study the mobility of small-amplitude solitons attached to moving defects which drag the solitons across a two-dimensional (2D) discrete nonlinear Schrödinger lattice. Findings are compared to the situation when a free small-amplitude 2D discrete soliton is kicked in a uniform lattice. In agreement with previously known results, after a period of transient motion the free soliton transforms into a localized mode pinned by the Peierls-Nabarro potential, irrespective of the initial velocity. However, the soliton attached to the moving defect can be dragged over an indefinitely long distance (including routes with abrupt turns and circular trajectories) virtually without losses, provided that the dragging velocity is smaller than a certain critical value. Collisions between solitons dragged by two defects in opposite directions are studied too. If the velocity is small enough, the collision leads to a spontaneous symmetry breaking, featuring fusion of two solitons into a single one, which remains attached to either of the two defects.

Mobility of high-power solitons in saturable nonlinear photonic lattices

We theoretically study the properties of one-dimensional nonlinear saturable photonic lattices exhibiting multiple mobility windows for stationary solutions. The effective energy barrier decreases to a minimum in those power regions where a new intermediate stationary solution appears. As an application, we investigate the dynamics of high-power Gaussian-like beams finding several regions where the light transport is enhanced.